# On Some New Paranormed Difference Sequence Spaces Defined by Orlicz Functions

Tripathy, Binod Chandra;Dutta, Hemen

• Accepted : 2009.07.09
• Published : 2010.03.31
• 28 6

#### Abstract

The main aim of this article is to introduce a new class of sequence spaces using the concept of n-norm and to investigate these spaces for some linear topological structures as well as examine these spaces with respect to derived (n-1)-norm. We use an Orlicz function, a bounded sequence of positive real numbers and some difference operators to construct these spaces so that they become more generalized and some other spaces can be derived under special cases. These investigations will enhance the acceptability of the notion of n-norm by giving a way to construct different sequence spaces with elements in n-normed spaces.

#### Keywords

n-norm;difference spaces;Orlicz function;paranorm;completeness

#### References

1. M. Et and R. Colak, On generalized difference sequence spaces, Soochow Jour. Math., 21(4)(1985), 147 - 169.
2. S. Gahler, 2-metrische Rume und ihre topologische Struktur, Math. Nachr., 28(1963), 115-148.
3. S. Gahler, Linear 2-normietre Rume, Math. Nachr., 28(1965), 1-43.
4. S. Gahler, Uber der Uniformisierbarkeit 2-metrische Rume, Math. Nachr., 28(1965), 235-244.
5. H. Gunawan, On n-Inner Product, n-Norms, and The Cauchy-Schwarz Inequality, Scientiae Mathematicae Japonicae, 5(2001), 47-54.
6. H. Gunawan, The space of p-summble sequences and its natural n- norm, Bull. Aust. Math. Soc., 64(1)(2001), 137-147. https://doi.org/10.1017/S0004972700019754
7. H. Gunawan and Mashadi, On finite Dimensional 2- normed spaces, Soochow J. of Math., 27(3)(2001), 631-639.
8. H. Gunawan and Mashadi M., On n- normed spaces, Int. J. Math. Math. Sci., 27(10)(2001), 631-639. https://doi.org/10.1155/S0161171201010675
9. H. Kizmaz, On certain sequence spaces, Canad. Math. Bull., 24(2)(1981), 169-176. https://doi.org/10.4153/CMB-1981-027-5
10. J. Lindenstrauss and L. Tzafriri, On Orlicz sequence spaces, Israel J. Math., 10(1971), 379-390. https://doi.org/10.1007/BF02771656
11. A. Misiak, n-inner product spaces, Math. Nachr., 140(1989), 299-319. https://doi.org/10.1002/mana.19891400121
12. Mursaleen, A. Khan and Qamaruddin, Difference sequence spaces defined by Orlicz functions, Demonstratio Mathematica, XXXII(1)(1999), 145-150.
13. S. D. Parasar and B. Choudhary, Sequence spaces defined by Orlicz functions, Indian J. Pure Appl. Math, 25(4)(1994), 419-428.
14. B. C. Tripathy and A. Esi, A new type of difference sequence spaces, International Journal of Sci. and Tech., 1(1)(2006), 11-14.
15. B .C. Tripathy, A. Esi and B. K. Tripathy, On a new type of generalized difference Cesaro Sequence spaces, Soochow J. Math., 31(3)(2005), 333-340.

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